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Recent research has pointed out the importance of the inequational exchange law (P * Q) ; (R * S) ≤ (P ; R) * (Q ; S) for concurrent processes. In particular, it has been shown that this law is equivalent to validity of the concurrency rule for Hoare triples. Unfortunately, the law does not hold in the relationally based setting of algebraic separation logic. However, we show that under mild conditions the reverse inequation (P ; R) * (Q ; S) ≤ (P * Q) ; (R * S) still holds there. Separating conjunction * in that calculus can be interpreted as true concurrency on disjointly accessed resources. From the reverse exchange law we derive slightly restricted but still reasonably useful variants of the concurrency rule. Moreover, using a corresponding definition of locality, we obtain also a variant of the frame rule. By this, the relational setting can also be applied for modular and concurrency reasoning. Finally, we present several variations of the approach to further interpret the results.

We use the by now well established setting of modal semirings to derive a modal algebra for Petri nets. It is based on a relation-algebraic calculus for separation logic that enables calculations of properties in a pointfree fashion and at an abstract level. Basically, we start from an earlier logical approach to Petri nets that in particular uses modal box and diamond operators for stating properties about the state space of such a net. We provide relational translations of the logical formulas which further allow the characterisation of general behaviour of transitions in an algebraic fashion. From the relational structure an algebra for frequently used properties of Petri nets is derived. In particular, we give connections to typical used assertion classes of separation logic. Moreover, we demonstrate applicability of the algebraic approach by calculations concerning a standard example of a mutex net.Walter Vogler and Bernhard Möller have now for about 22 years been working at the Institute for Informatics at the University of Augsburg in close neighbourhood. Both were and are very much interested in formal semantics, although in different areas: Walter in concurrent, Bernhard in sequential systems. Still there were many debates whether or how their fields of work could have concrete common touchpoints, in particular, since both use algebraic concepts to a smaller or larger extent. Since in recent years Bernhard also started looking at the concurrent side, we felt that Walter's upcoming jubilee was the right point in time to try and construct a bridge (or at least a gangplank) between the two fields. Therefore it is our great pleasure to dedicate this paper to Walter Vogler on the occasion of his 60th birthday, also with sincere thanks for his friendship and the pleasant collaboration. May he continue to throw out his nets to bring in an ample harvest of impressive results! This research was partially funded by the DFG project MO 690/9-1 AlgSep-Algebraic Calculi for Separation Logic.

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